Section 3.5
Derivatives of Trig Functions
Trigonometric functions model periodic motion, so their derivatives are also periodic (and cyclical).
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The Six Basic Derivatives
You must memorize these. Notice the pattern: all "co-" functions (cosine, cosecant, cotangent) have negative derivatives.
Primary Functions
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"Co-" Functions (Negative)
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Worked Example
Differentiate .
Step 1: Identify the Rule
This is a product of two functions: and .
Use the Product Rule: .
Step 2: Differentiate Components
Step 3: Assemble
Usually, we factor out common terms if convenient, but this form is correct.
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More Examples
Product Rule with Trig
Differentiate .
Identify components:
Apply Product Rule :
Quotient Rule with Trig
Differentiate .
Identify Hi and Lo:
Apply Quotient Rule:
Simplify numerator:
Use identity :
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Interactive Visualization
Sine and its Derivative
Observe the beautiful relationship: The derivative of sin(x) is cos(x). Notice that when sine has a peak (slope 0), cosine crosses the x-axis (value 0).
Original Function f(x)
Tangent Line
Derivative Function f'(x)
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Practice Quiz
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